If you want a DR that will handle turns, the NT05 and MT whave a soft sidewall, and woulnt turn well. you can have one that turns, or one that goes straight- Im not sure about the Toyos, but the only DR that can handle and go straight would be the 555R, which is quite hard compared to the above.
That said, if there are guys running 10.80s or better on vintage 1960's bias ply, there is no reason for you to need a monstrously soft drag radial to hook in 1st and 2nd gear. the 555R would do it fine, and is a compromise to both your needs. But like an all season tire, it wont excel greatly in any one area.